Explicit Analysis of Sheet Metal Forming Processes Using Solid-Shell Elements

Abstract

To simulate sheet metal forming processes precisely, an in-house dynamic explicit code was developed to apply a new solid-shell element to sheet metal forming analyses, with a corotational coordinate system utilized to simplify the nonlinearity and to integrate the element with anisotropic constitutive laws. The enhancing parameter of the solid-shell element, implemented to circumvent the volumetric and thickness locking phenomena, was condensed into an explicit form. To avoid the rank deficiency, a modified physical stabilization involving the B-bar method and reconstruction of transverse shear components was adopted. For computational efficiency of the solid-shell element in numerical applications, an adaptive mesh subdivision scheme was developed, with element geometry and contact condition taken as subdivision criteria. To accurately capture the anisotropic behavior of sheet metals, material models with three different anisotropic yield functions were incorporated. Several numerical examples were carried out to validate the accuracy of the proposed element and the efficiency of the adaptive mesh subdivision.

1. Introduction

Recently, solid-shell elements have been a focus of research on modeling thin-walled structures [1,2]. As compared with shell elements which have been widely used for several decades, solid-shell elements are more suitable for double-sided contact situations such as sheet metal forming due to the existence of nodes at the upper and lower surfaces. In addition, without nodal rotational degrees of freedom, the kinetic and geometric descriptions of solid-shell elements are simpler than the descriptions of shell elements. A further advantage is that general 3D constitutive models can be used together with solid-shell elements, and in this way, the thickness variation can be precisely described, which makes solid-shell elements highly desirable when a workpiece with varying or large thickness is analyzed.

However, solid-shell elements have been plagued by various locking problems, and therefore the elements provide poor results. For example, when dealing with nearly incompressible situations, i.e., Poisson ratio approaching 0.5, such as rubber-like materials and metal plasticity, the artificial constraint at integration points may lead to volumetric locking. Another well-known locking problem associated with Poisson ratio is known as thickness locking or Poisson locking. It usually happens in out-of-plane bending analyses in which the displacement is assumed to vary linearly along the thickness direction, and as a result, a constant strain field along the thickness direction is obtained. Nevertheless, because the Poisson ratio is not equal to zero, the coupling between the normal thickness strain and the in-plane strains makes the normal thickness strain vary linearly in the thickness direction. In bending deformations of shell-type structures, transverse shear locking is also a severe problem, which appears when the element thickness is extremely small. Finally, in modeling curved structures with solid-shell elements, trapezoidal locking, also named as curvature locking, may take place when the element edges in the thickness direction are not perpendicular to the element mid-surface.

There have been significant efforts made to solve the abovementioned locking problems, among which, reduced integration and selective reduced integration are commonly adopted to treat volumetric locking and transverse shear locking [3]. Nevertheless, due to the rank deficiency caused by reduced integration, some stabilization procedures are necessary to avoid the appearance of hourglass modes. The enhanced assumed strain (EAS) method, which was proposed by Simo and Rifai [4], has been widely used in many studies to deal with volumetric locking, thickness locking, transverse shear locking, etc. For example, in a study by Su et al. [5], an in-plane reduced integration solid-shell element was proposed, with seven internal parameters used to enhance the strain field. In addition, in the solid-shell element presented by Li et al. [6], six internal parameters were adopted to attenuate transverse shear locking, with one other parameter used for volumetric and thickness locking. Although the EAS method is attractive for dealing with various locking, the extra internal parameters introduced by this approach will inevitably lead to an increase in numerical computations. To overcome the computational inefficiency of the EAS method, there is a need to introduce as few internal parameters as possible. As compared with the EAS method, the assumed natural strain (ANS) method is considered to be a more efficient technique to treat transverse shear and trapezoidal locking. Therefore, numerous solid-shell elements have been developed by combining the EAS and ANS such as the work of Mostafa et al. [7,8], Schwarze et al. [9], Cardoso et al. [10], and Norachan et al. [11].

Even though many proposed solid-shell elements show excellent performance in classical thin-walled tests, there are fewer applications of solid-shell element in practical engineering examples such as sheet metal forming as compared with shell elements. Via solid-shell elements, Schwarze et al. [12,13] and Parente et al. [14] analyzed the S-rail forming problem and the deep drawing of a cylindrical cup with anisotropy. In Schwarze et al.’s study [12], the deep drawing of a square cup was also simulated. In a study conducted by Wang et al. [15], several typical benchmark problems including a single point incremental sheet metal forming and spring back of a U-shaped component were employed to assess the capability of their proposed solid-shell element in sheet metal forming. The behavior of a solid-shell element in magnetic pulse forming was investigated by Mahmoud et al. [16]. Additionally, a solid-shell element was incorporated into an in-house program by Li et al. [6] to simulate the roll forming of a U-channel component.

The underdevelopment of solid-shell elements in the field of sheet metal forming may result from the technology adopted in the element formulation. As claimed in [17], there are only a few contributions using EAS in combination with explicit time integration. The EAS is based on the introduction of additional degrees of freedom at the element level, which can be condensed using local equation calculation and leads to exactly accurate element formulations in the context of implicit time integration methods. However, the application of EAS in an explicit framework will lead to an increase in the computational effort, especially in elements with many EAS parameters. As compared with implicit time integration schemes, explicit time integration schemes are more suitable for complex examples involving material nonlinearity, geometric nonlinearity, and changeable contact constraints between tools and workpiece. Thus, to investigate the application of solid-shell element in an explicit framework, the effect of the EAS to computational efficiency should be considered.

Computational effort of an explicit simulation is directly proportional to the number of mesh nodes. To decrease the total node number and to improve the computational efficiency dramatically, an adaptive meshing method for shell elements has been widely used in some well-known finite element analysis software for sheet metal forming such as Dynaform (ETA) [18] and Autoform (AutoForm Engineering GmbH, Pfäffikon, Switzerland) [19]. Nevertheless, this method has not been extensively studied in the field of solid-shell element. In a study by Xie et al. [20], a curvature-based mesh subdivision was proposed to avoid the non-physical interpenetration of element surfaces in fabric drape simulations. In addition, in the literature published by Sena et al. [21], an adaptive remeshing method developed for shell elements was applied to single point incremental forming simulations with a solid-shell element.

In our previous study [22], a solid-shell element was proposed for geometrically linear problems. To investigate the applicability of the proposed element in sheet metal forming analyses, an explicit in-house code with the nonlinear formulation of the element is developed in this study. The remainder of this paper is organized as follows: The solid-shell formulation is described in Section 2. After a brief introduction of the dynamic explicit algorithm in Section 3, an adaptive mesh subdivision scheme for the solid-shell element is proposed in Section 4. For the sake of sheet metal forming simulation with anisotropy, material models based on three yield functions, i.e., Hill48 [23], Yld91 [24], and Yld2004-18p [25], are summarized in Section 5. Finally, several numerical examples are presented to verify the accuracy and efficiency of the proposed element in sheet metal forming.

2. Solid-Shell Formulation

2.1. Geometrics and Kinematic

The isoparametric eight-node hexahedral solid-shell is shown in Figure 1. With ξ, η, and ζ representing the natural coordinates, only one integration point is used in the ξη-plane, and multiple integration points distribute along the ζ-axis. Via this special integration scheme, the element can accurately describe the stress variation in the thickness direction.

Each node of the element has three translational degrees of freedom and no rotational degree of freedom. The position vector $x={\{x,y,z\}}^{\mathrm{T}}$ and displacement vector $u={\{u,v,w\}}^{\mathrm{T}}$ at any point of the isoparametric domain can be obtained by the following interpolation functions:

$$x={\displaystyle \sum _{i=1}^8{N}_i(\xi ,\eta ,\zeta )x_i}=Nc \mathrm{and} u={\displaystyle \sum _{i=1}^8{N}_i(\xi ,\eta ,\zeta )u_i}=Nd$$

(1)

where $N=\left[\begin{array}{cccc}{N}_1{I}_3& N_2{I}_3& \dots & N_8{I}_3\end{array}\right]$ with $N_i(\xi ,\eta ,\zeta )=\frac{1}{8}(1+{\xi }_i\xi )(1+{\eta }_i\eta )(1+{\zeta }_i\zeta )$ denote the trilinear shape function at node I, $c$ is the nodal coordinate vector, and $d$ is the nodal displacement vector.

Considering the nonorthogonality of the natural frame, a corotational coordinate system is established at the element centroid to simplify the geometric and material nonlinearities and describe the strain and stress more objectively, with its unit vectors defined as:

$$\begin{array}{l}{r}_1={\displaystyle \sum _{i=1}^8{\xi }_i{x}_i} r_1=\frac{r_1}{\left|r_1\right|}\\ r_3=r_1\times {\displaystyle \sum _{i=1}^8{\eta }_i{x}_i} r_3=\frac{r_3}{\left|r_3\right|}\\ r_2=r_3\times r_1\end{array}$$

(2)

Thus, the natural frame and the local coordinate system are related by:

$$t={\left[\begin{array}{ccc}{r}_1& r_2& r_3\end{array}\right]}^{\mathrm{T}}{J}^{-1}$$

(3)

where J is the conventional Jacobian.

When a sheet metal forming analysis is conducted in the explicit framework, the deformation in a time step is small enough. Thus, the increment of the covariant strain ${\overline{\mathsf{\epsilon }}}_u$ in time step $n\to n+1$ can be computed from the displacement ${}_n{}^{n+1}d$ by:

$${}_n{}^{n+1}\overline{\mathsf{\epsilon }}{}_u={\overline{B}}_u{}_n{}^{n+1}d$$

(4)

The strain increment ${}_n{}^{n+1}\tilde{\mathsf{\epsilon }}{}_u$ in the corotational coordinate system is connected to the covariant strain increment ${}_n{}^{n+1}\overline{\mathsf{\epsilon }}{}_u$ via ${}_n{}^{n+1}\tilde{\mathsf{\epsilon }}{}_u=T{}_n{}^{n+1}\overline{\mathsf{\epsilon }}{}_u$, where the 6 × 6 transformation matrix T is expressed by the terms of matrix t, as follows:

$$T=\left[\begin{array}{cccccc}{t}_{11}^{}{t}_{11}^{}& t_{12}^{}{t}_{12}^{}& t_{13}^{}{t}_{13}^{}& t_{11}^{}{t}_{12}^{}& t_{12}^{}{t}_{13}^{}& t_{13}^{}{t}_{11}^{}\\ t_{21}^{}{t}_{21}^{}& t_{22}^{}{t}_{22}^{}& t_{23}^{}{t}_{23}^{}& t_{21}^{}{t}_{22}^{}& t_{22}^{}{t}_{23}^{}& t_{23}^{}{t}_{21}^{}\\ t_{31}^{}{t}_{31}^{}& t_{32}^{}{t}_{32}^{}& t_{33}^{}{t}_{33}^{}& t_{31}^{}{t}_{32}^{}& t_{32}^{}{t}_{33}^{}& t_{33}^{}{t}_{31}^{}\\ 2t_{11}^{}{t}_{21}^{}& 2t_{12}^{}{t}_{22}^{}& 2t_{13}^{}{t}_{23}^{}& t_{11}^{}{t}_{22}^{}+t_{12}^{}{t}_{21}^{}& t_{12}^{}{t}_{23}^{}+t_{13}^{}{t}_{22}^{}& t_{13}^{}{t}_{21}^{}+t_{11}^{}{t}_{32}^{}\\ 2t_{21}^{}{t}_{31}^{}& 2t_{22}^{}{t}_{32}^{}& 2t_{23}^{}{t}_{33}^{}& t_{21}^{}{t}_{32}^{}+J_{22}^{}{t}_{31}^{}& t_{22}^{}{t}_{33}^{}+t_{23}^{}{t}_{32}^{}& t_{23}^{}{t}_{31}^{}+t_{21}^{}{t}_{33}^{}\\ 2t_{31}^{}{t}_{11}^{}& 2t_{32}^{}{t}_{12}^{}& 2t_{33}^{}{t}_{13}^{}& t_{31}^{}{t}_{12}^{}+t_{32}^{}{t}_{11}^{}& t_{32}^{}{t}_{13}^{}+t_{33}^{}{t}_{12}^{}& t_{33}^{}{t}_{11}^{}+t_{31}^{}{t}_{13}^{}\end{array}\right]$$

(5)

Therefore, we have:

$${}_n{}^{n+1}\tilde{\mathsf{\epsilon }}{}_u={\tilde{B}}_u{}_n{}^{n+1}d$$

(6)

with

$${\tilde{B}}_u=T{\overline{B}}_u$$

(7)

2.2. EAS Method

The EAS method is employed to overcome the volumetric and thickness locking problems of the solid-shell element, with the strain field enriched as follows:

$${}_n{}^{n+1}\overline{\mathsf{\epsilon }}={}_n{}^{n+1}\overline{\mathsf{\epsilon }}{}_u+{}_n{}^{n+1}\overline{\mathsf{\epsilon }}{}_{\alpha }$$

(8)

The increment of the enhancing strain can be written in a similar form with Equation (4), as:

$${}_n{}^{n+1}\overline{\mathsf{\epsilon }}{}_{\alpha }={\overline{B}}_{\alpha }{}_n{}^{n+1}\mathsf{\alpha }$$

(9)

with ${\overline{B}}_{\alpha }^{}={\left[\begin{array}{cccccc}0& 0& \zeta & 0& 0& 0\end{array}\right]}^{\mathrm{T}}$. For computational efficiency, only one internal variable is introduced in each element, i.e., $\mathsf{\alpha }=\left\{{\alpha }_1\right\}$. In this way, an enhancing strain field linear to $\zeta$ is added to the normal thickness component of ${}_n{}^{n+1}\overline{\mathsf{\epsilon }}{}_u$. With $T^0$ denoting the transformation matrix T evaluated at the element center $(\xi =0,\eta =0,\zeta =0)$, ${}_n{}^{n+1}\overline{\mathsf{\epsilon }}{}_{\alpha }$ can be transformed into the corotational coordinate system by ${}_n{}^{n+1}\tilde{\mathsf{\epsilon }}{}_{\alpha }=T^0{}_n{}^{n+1}\overline{\mathsf{\epsilon }}{}_{\alpha }$.

The Veubeke–Hu–Washizu variational principle in large deformations can be expressed by:

$$\delta \mathsf{\Pi }(d,\tilde{\mathsf{\epsilon }},\widehat{\mathsf{\sigma }})={\displaystyle {\mathrm{\int }}_{V^e}\delta {\tilde{\mathsf{\epsilon }}}^{\mathrm{T}}\tilde{\mathsf{\sigma }}}\mathrm{d}V+\delta {\displaystyle {\mathrm{\int }}_{V^e}{\widehat{\mathsf{\sigma }}}^{\mathrm{T}}}(\nabla d-\tilde{\mathsf{\epsilon }})\mathrm{d}V+\delta {\mathsf{\Pi }}_1-\delta {\mathsf{\Pi }}_2=0$$

(10)

with

$$\left\{\begin{array}{l}\delta {\mathsf{\Pi }}_1=\delta d^{\mathrm{T}}{\displaystyle {\mathrm{\int }}_{V^e}\rho N^{\mathrm{T}}N}\mathrm{d}V\ddot{d}=\delta d^{\mathrm{T}}M\ddot{d}\\ \delta {\mathsf{\Pi }}_2=\delta d^{\mathrm{T}}{F}^{\mathrm{ext}}\end{array}\right.$$

(11)

where $\tilde{\mathsf{\epsilon }}$ denotes the assumed strain field, $\tilde{\mathsf{\sigma }}$ the assumed stress field, $M$ the element mass matrix, and $F^{\mathrm{ext}}$ the external nodal force vector. Due to the orthogonality condition as follows:

$${\displaystyle {\int }_{V^e}{\widehat{\mathsf{\sigma }}}^{\mathrm{T}}}(\nabla d-\tilde{\mathsf{\epsilon }})\mathrm{d}V=0$$

(12)

the three-field variational function is simplified to:

$$\delta \mathsf{\Pi }(d,\tilde{\mathsf{\epsilon }})={\displaystyle {\mathrm{\int }}_{V^e}\delta {\tilde{\mathsf{\epsilon }}}^{\mathrm{T}}\tilde{\mathsf{\sigma }}}\mathrm{d}V+\delta d^{\mathrm{T}}M\ddot{d}-\delta d^{\mathrm{T}}{F}^{\mathrm{ext}}=0$$

(13)

Considering that the assumed strain $\tilde{\mathsf{\epsilon }}$ consists of compatible and incompatible parts, we have:

$$\delta d^{\mathrm{T}}(F^{\mathrm{int}}+M\ddot{d}-F^{\mathrm{ext}})=0$$

(14)

$$\delta {\mathsf{\alpha }}^{\mathrm{T}}{F}_{\alpha }^{\mathrm{int}}=0$$

(15)

where $F^{\mathrm{int}}={\displaystyle {\mathrm{\int }}_{V^e}{\tilde{B}}_u{}^{\mathrm{T}}\tilde{\mathsf{\sigma }}\mathrm{d}V}$ and $F_{\alpha }^{\mathrm{int}}={\displaystyle {\mathrm{\int }}_{V^e}{\tilde{B}}_{\alpha }{}^{\mathrm{T}}\tilde{\mathsf{\sigma }}\mathrm{d}V}$ are internal force vectors.

The compatible degrees of freedom $d$ can be calculated by the central difference method, which is described in the next section. Concerning the internal degree of freedom $\mathsf{\alpha }$, the linearization of $F_{\alpha }^{\mathrm{int}}(d,\mathsf{\alpha })=0$ in the time step results in:

$$\tilde{K}{(}^{n}d,{}^{\mathit{n}}\mathsf{\alpha }^{\mathit{k}})({}^n\mathsf{\alpha }^{k+1}-{}^n\mathsf{\alpha }^k)=-F_{\alpha }^{\mathrm{int}}{(}^{n}d,{}^n\mathsf{\alpha }^k)$$

(16)

where $k$ denotes the iterative times of ${}_{}{}^n\mathsf{\alpha }$, ${}^n\mathsf{\alpha }^{k=0}={}^{n-1}\mathsf{\alpha }$, and $\tilde{K}{(}^{n}d,{}^n\mathsf{\alpha }^k)={\left.\frac{\partial F_{\alpha }^{\mathrm{int}}}{\partial \mathsf{\alpha }}\right|}_{{}^n\mathsf{\alpha }={}^n\mathsf{\alpha }^k}={\left.{{\displaystyle {\int }_{V^e}{\tilde{B}}_{\alpha }{}^{\mathrm{T}}D}}^{\mathrm{ep}}{\tilde{B}}_{\alpha }\mathrm{d}V\right|}_{{}^n\mathsf{\alpha }={}^n\mathsf{\alpha }^k}$. Due to the small time step of the explicit time integration, one single iteration is usually enough to obtain the correct result [17], and therefore the explicit form:

$${}^n\mathsf{\alpha }={}^{n-1}\mathsf{\alpha }-{\tilde{K}}^{-1}{(}^{n}d,{}^{n-1}\mathsf{\alpha })F_{\alpha }^{\mathrm{int}}{(}^{n}d,{}^{n-1}\mathsf{\alpha })$$

(17)

can be adopted. With only one EAS parameter introduced in the element, $\tilde{K}$ is essentially a scalar. Thus, no matrix inversion is involved in Equation (17), and ${}_{}{}^n\mathsf{\alpha }$ can be worked out efficiently.

2.3. ANS Method

To avoid trapezoidal locking and transverse shear locking of the solid-shell element, the ANS method is employed in evaluations of the normal thickness term and the transverse shear terms of ${}_n{}^{n+1}\overline{\mathsf{\epsilon }}$ [22].

2.4. Hourglass Stabilization

Via Equation (7), we obtain ${\tilde{B}}_u$ in the following polynomial form:

$${\tilde{B}}_u=\underset{{\tilde{B}}^{\ast }}{\underbrace{{\tilde{B}}^0+{\tilde{B}}^{\zeta }\zeta }}+\underset{{\tilde{B}}^{\mathrm{H}}}{\underbrace{{\tilde{B}}^{\xi }\xi +{\tilde{B}}^{\eta }\eta +{\tilde{B}}^{\eta \zeta }\eta \zeta +{\tilde{B}}^{\zeta \xi }\zeta \xi }}$$

(18)

It should be noted that the term ${\tilde{B}}^{\xi \eta }$ is not considered to be in ${\tilde{B}}^{\mathrm{H}}$ matrix since it is not needed in correcting the rank deficiency [26].

Locking problems must be controlled in the stabilization procedure. To overcome the volumetric locking phenomenon, the B-bar method [3,27] is utilized with the ${\tilde{B}}^{\mathrm{H}}$ matrix split into deviatoric and dilatational parts, and the dilatational part is integrated at the element center, as follows:

$${\tilde{B}}^{\mathrm{H}}(\xi ,\eta ,\zeta )={}^{\mathrm{dev}}\tilde{\mathbf{B}}^{\mathrm{H}}(\xi ,\eta ,\zeta )+{}^{\mathrm{dil}}\tilde{\mathbf{B}}^{\mathrm{H}}(0,0,0)$$

(19)

Both ${}^{\mathrm{dev}}\tilde{\mathbf{B}}^{\mathrm{H}}$ and ${}^{\mathrm{dil}}\tilde{\mathbf{B}}^{\mathrm{H}}$ can be expanded according to the expansion of ${\tilde{B}}^{\mathrm{H}}$ in Equation (18). Without constant term in the expansion, the dilatational part is equal to zero. As a result, we have:

$${\tilde{\mathbf{B}}}^{\mathrm{H}}(\xi ,\eta ,\zeta )={}^{\mathrm{dev}}\tilde{\mathbf{B}}^{\mathrm{H}}(\xi ,\eta ,\zeta )$$

(20)

It has been demonstrated that physical stabilization probably makes element behave overly stiff in bending-dominated situations, especially when the element width-to-thickness ratio tends to zero [3,10,28]. Aiming at improving the performance of the proposed element, the transverse shear terms of ${\tilde{B}}^{\mathrm{H}}$ are reconstructed by:

$$\begin{array}{l}{\tilde{B}}_{yz}^{\mathrm{H}}={\tilde{B}}_{yz}^{\eta \zeta }\eta \zeta +{\tilde{B}}_{yz}^{\zeta \xi }\zeta \xi \\ {\tilde{B}}_{zx}^{\mathrm{H}}={\tilde{B}}_{zx}^{\eta \zeta }\eta \zeta +{\tilde{B}}_{zx}^{\zeta \xi }\zeta \xi \end{array}$$

(21)

in which the linear parts contributing to the excessive hourglass stiffness are dropped out.

Via the ${\tilde{B}}^{\mathrm{H}}$ matrix, the increment of the hourglass strain tensor can be obtained in the following way:

$${}_n{}^{n+1}\tilde{\mathsf{\epsilon }}^{\mathrm{H}}={}_n{}^{n+1}\tilde{\mathsf{\epsilon }}^{\xi }\xi +{}_n{}^{n+1}\tilde{\mathsf{\epsilon }}^{\eta }\eta +{}_n{}^{n+1}\tilde{\mathsf{\epsilon }}^{\eta \zeta }\eta \zeta +{}_n{}^{n+1}\tilde{\mathsf{\epsilon }}^{\zeta \xi }\zeta \xi$$

(22)

Accordingly, the hourglass stress tensor can be written in the form ${}_n{}^{n+1}\tilde{\mathsf{\sigma }}^{\mathrm{H}}={}_n{}^{n+1}\tilde{\mathsf{\sigma }}^{\xi }\xi +{}_n{}^{n+1}\tilde{\mathsf{\sigma }}^{\eta }\eta +{}_n{}^{n+1}\tilde{\mathsf{\sigma }}^{\eta \zeta }\eta \zeta +{}_n{}^{n+1}\tilde{\mathsf{\sigma }}^{\zeta \xi }\zeta \xi$ with the terms in the polynomial calculated by:

$$\begin{array}{ll}{}_n{}^{n+1}\tilde{\mathsf{\sigma }}^{\xi }=\frac{E^t}{E}{D}^{\mathrm{e}}{}_n{}^{n+1}\tilde{\mathsf{\epsilon }}^{\xi }& {}_n{}^{n+1}\tilde{\mathsf{\sigma }}^{\eta }=\frac{E^t}{E}{D}^{\mathrm{e}}{}_n{}^{n+1}\tilde{\mathsf{\epsilon }}^{\eta }\\ {}_n{}^{n+1}\tilde{\mathsf{\sigma }}^{\eta \zeta }=\frac{E^t}{E}{D}^{\mathrm{e}}{}_n{}^{n+1}\tilde{\mathsf{\epsilon }}^{\eta \zeta }& {}_n{}^{n+1}\tilde{\mathsf{\sigma }}^{\zeta \xi }=\frac{E^t}{E}{D}^{\mathrm{e}}{}_n{}^{n+1}\tilde{\mathsf{\epsilon }}^{\zeta \xi }\end{array}$$

(23)

where $E$ represents the Young’s modulus, $E^t$ is the tangent modulus evaluating at the element center, and $D^{\mathrm{e}}$ denotes the full 3D elastic constitutive model.

With the hourglass stabilization, the $F^{\mathrm{int}}$ in Equation (14) can be rewritten as:

$$F^{\mathrm{int}}=\underset{F^{*}}{\underbrace{{\displaystyle {\mathrm{\int }}_{V^e}{\left({\tilde{B}}^{*}\right)}^{\mathrm{T}}{\tilde{\mathsf{\sigma }}}^{*}\mathrm{d}V}}}+\underset{F^{\mathrm{H}}}{\underbrace{{\displaystyle {\mathrm{\int }}_{V^e}{\left({\tilde{B}}^{\mathrm{H}}\right)}^{\mathrm{T}}{\tilde{\mathsf{\sigma }}}^{\mathrm{H}}\mathrm{d}V}}}+\underset{=0}{\underbrace{{\displaystyle {\mathrm{\int }}_{V^e}{\left({\tilde{B}}^{*}\right)}^{\mathrm{T}}{\tilde{\mathsf{\sigma }}}^{\mathrm{H}}\mathrm{d}V}}}+\underset{=0}{\underbrace{{\displaystyle {\mathrm{\int }}_{V^e}{\left({\tilde{B}}^{\mathrm{H}}\right)}^{\mathrm{T}}{\tilde{\mathsf{\sigma }}}^{*}\mathrm{d}V}}}.$$

(24)

Since the hourglass stress tensor ${\tilde{\mathsf{\sigma }}}^{\mathrm{H}}$ is accumulated by ${}^{n+1}\tilde{\mathsf{\sigma }}^{\mathrm{H}}={}^n\tilde{\mathsf{\sigma }}^{\mathrm{H}}+{}_n{}^{n+1}\tilde{\mathsf{\sigma }}^{\mathrm{H}}$, it can also be expanded into linear and bilinear terms. By substituting ${\tilde{\mathsf{\sigma }}}^{\mathrm{H}}$ into $F^{\mathrm{H}}={\displaystyle {\mathrm{\int }}_{V^e}{\left({\tilde{B}}^{\mathrm{H}}\right)}^{\mathrm{T}}{\tilde{\mathsf{\sigma }}}^{\mathrm{H}}\mathrm{d}V}$ and considering the integrals

$$\begin{array}{l}{\displaystyle {\mathrm{\int }}_{-1}^1{\displaystyle {\mathrm{\int }}_{-1}^1{\displaystyle {\mathrm{\int }}_{-1}^1{\xi }^2\mathrm{d}\xi \mathrm{d}\eta \mathrm{d}\zeta }}}={\displaystyle {\mathrm{\int }}_{-1}^1{\displaystyle {\mathrm{\int }}_{-1}^1{\displaystyle {\mathrm{\int }}_{-1}^1{\eta }^2\mathrm{d}\xi \mathrm{d}\eta \mathrm{d}\zeta }}}=\frac{8}{3}\\ {\displaystyle {\mathrm{\int }}_{-1}^1{\displaystyle {\mathrm{\int }}_{-1}^1{\displaystyle {\mathrm{\int }}_{-1}^1{\eta }^2{\xi }^2\mathrm{d}\xi \mathrm{d}\eta \mathrm{d}\zeta }}}={\displaystyle {\mathrm{\int }}_{-1}^1{\displaystyle {\mathrm{\int }}_{-1}^1{\displaystyle {\mathrm{\int }}_{-1}^1{\zeta }^2{\xi }^2\mathrm{d}\xi \mathrm{d}\eta \mathrm{d}\zeta }}}=\frac{8}{9}\end{array}$$

(25)

over the natural coordinates,

$$F^{\mathrm{H}}=\frac{8}{3}\left|J^0\right|{\tilde{B}}^{\xi \mathrm{T}}{\tilde{\mathsf{\sigma }}}^{\xi }+\frac{8}{3}\left|J^0\right|{\tilde{B}}^{\eta \mathrm{T}}{\tilde{\mathsf{\sigma }}}^{\eta }+\frac{8}{9}\left|J^0\right|{\tilde{B}}^{\eta \zeta \mathrm{T}}{\tilde{\mathsf{\sigma }}}^{\eta \zeta }+\frac{8}{9}\left|J^0\right|{\tilde{B}}^{\zeta \xi \mathrm{T}}{\tilde{\mathsf{\sigma }}}^{\zeta \xi }$$

(26)

can be analytically calculated, with $\left|J^0\right|$ being the determinant of the Jacobian matrix evaluated at the element center.

3. Central Difference Scheme and Time Step Estimation

According to Equations (14) and (24), the equilibrium equation for the dynamic analysis of sheet metal forming is expressed in the form:

$${\displaystyle \sum _{\mathrm{Ass}}M\ddot{d}}={\displaystyle \sum _{\mathrm{Ass}}{F}^{\mathrm{ext}}}-{\displaystyle \sum _{\mathrm{Ass}}(F^{*}+F^{\mathrm{H}})}$$

(27)

with ${\displaystyle \sum _{\mathrm{Ass}}•}$ denoting the assembly of matrixes or vectors in the global coordinate system. The mass matrix ${\displaystyle \sum _{\mathrm{Ass}}M}$ in Equation (27) is diagonalized. In this way, the assembled equilibrium equation can be uncoupled into:

$$m_j{\ddot{d}}_j=f_j^{\mathrm{ext}}-(f_j^{*}+f_j^{\mathrm{H}})$$

(28)

leading to more convenient and economic computational process. By defining $\beta =\Delta t_n/\Delta t_{n-1}$ and substituting the following central difference equations:

$$\begin{array}{l}{\dot{d}}_j^n=\frac{\beta }{1+\beta }{\dot{d}}_j^{n+1/2}+\frac{1}{1+\beta }{\dot{d}}_j^{n-1/2}\\ {\ddot{d}}_j^n=\frac{2}{(1+\beta )\Delta t_{n-1}}({\dot{d}}_j^{n+1/2}-{\dot{d}}_j^{n-1/2})\end{array}$$

(29)

into Equation (28), we have:

$${\dot{d}}_j^{n+1/2}={\dot{d}}_j^{n-1/2}+\frac{(1+\beta )\Delta t_{n-1}}{2}\frac{f_j^{\mathrm{ext}}{}^n-(f_j^{*}{}^n+f_j^{\mathrm{H}}{}^n)}{m_j}$$

(30)

Thus, the nodal velocity in time step $n\to n+1$ is solved explicitly. With $d_j^{n+1}=d_j^n+{\dot{d}}_j^{n+1/2}\cdot \Delta t_n$, the nodal displacement at $t=t_{n+1}$ can be obtained efficiently.

The central difference scheme is conditionally stable, and the time step $\Delta t$ should be less than a critical value $\Delta t^{\mathrm{cr}}$, that is:

$$\Delta t<\Delta t^{\mathrm{cr}}=\min \frac{l}{c}.$$

(31)

where $l$ represents the characteristic element size, and $c$ is the wave propagation speed in this element [29]. Such a critical value is calculated individually for each element, and the minimum value is selected as the time step to guarantee the computational stability.

4. Adaptive Mesh Subdivision

With finer mesh, the material flowing behavior in complex sheet metal forming processes can be captured more accurately. However, a finer mesh will inevitability lead to higher computational cost. To attain an ideal balance between accuracy and efficiency, an adaptive mesh subdivision scheme is implemented for the proposed solid-shell element. In simulations with the adaptive mesh subdivision, the entire workpiece is originally discretized by coarse mesh. As the workpiece deforms, the element that meets a criterion will be subdivided automatically. Considering that in most analyses the critical time step is determined by the element thickness, the mesh subdivision is only carried out in the element plane, as shown in Figure 2. Thus, the element thickness is insensitive to the subdivision. For higher computational efficiency, a multilevel subdivision can be adopted.

Finer mesh is necessary in complex-deformation areas, and complex deformation may lead to element distortion. For this reason, two geometry-based criteria are employed in the adaptive mesh subdivision. Given that the subdivision level of the initial coarse element is 0, and the user-defined highest subdivision level is $L^{\max }$, an element with level L is subdivided if its maximum warping angle is greater than the critical value, that is:

$$\vartheta =\max \left({\vartheta }^{AC},{\vartheta }^{BD}\right)>{\vartheta }_{\kappa }^{cr}(\kappa =L<L^{\max })$$

(32)

As shown in Figure 3, the warping angle ${\vartheta }^{AC}$ is the angle between the triangle ABC and the triangle ACD, and the warping angle ${\vartheta }^{BD}$ is the angle between the triangle ABD and the triangle BCD. The points A, B, C, and D indicate the midpoints of the four element edges in the thickness direction.

As exhibited in Figure 4, element E₁ and E₂ are adjacent, $n_1^{\mathrm{upper}}$ perpendicular to vectors V₁₃ and V₂₄ is defined as the normal vector of the upper surface of E₁, and $n_1^{\mathrm{lower}}$ perpendicular to vectors V₅₇ and V₆₈ is defined as the normal vector of the lower surface of E₁. Correspondingly, normal vectors $n_2^{\mathrm{upper}}$ and $n_2^{\mathrm{lower}}$ of element E₂ can be obtained in the same way. The angle between $n_1^{\mathrm{upper}}$ and $n_2^{\mathrm{upper}}$ is denoted as ${\theta }^{\mathrm{upper}}$, and the angle between $n_1^{\mathrm{lower}}$ and $n_2^{\mathrm{lower}}$ is ${\theta }^{\mathrm{lower}}$. For two adjacent elements with subdivision level $L_1$ and $L_2$, when the angle between them is greater than the critical value, that is:

$$\theta =\max \left({\theta }^{\mathrm{upper}},{\theta }^{\mathrm{lower}}\right)>{\theta }_{\kappa }^{cr}(\kappa =\min (L_1,L_2)<L^{\max })$$

(33)

the element with its subdivision level less than $L^{\max }$ will be subdivided. ${\theta }^{\mathrm{upper}}$ and ${\theta }^{\mathrm{lower}}$ denote the angle at the upper and lower surface, respectively. The critical values ${\vartheta }_{\kappa }^{cr}$ and ${\theta }_{\kappa }^{cr}$ are two sets of preset parameters increasing with the subdivision level $\kappa$.

Despite the overall excellent performance, the geometry-based adaptive subdivision may give rise to nodal displacement oscillation. This is because, in the developed finite element code, the master-slave contact algorithm is adopted, in which the penetration of slave (workpiece) node into the master (tool) surface is not allowed. However, when a newly inserted node produced by the mesh subdivision is located on the back side of the tool surface, as shown in Figure 5, the initial penetration will bring about a force acting on the newly inserted node. If the initial penetration is large enough, the large and sudden force will probably lead to excessive dynamic oscillation. To avoid this problem, even though Equations (32) and (33) are not satisfied, the workpiece element is subdivided once a tool node penetrates it and the penetration distance is greater than the critical value, which is set as 3% of the workpiece thickness in this paper.

To ensure the simulation accuracy and robustness, the subdivision process is forbidden to produce adjacent elements with their subdivision level difference larger than 1. In addition, to get rid of gaps or overlaps between two adjacent elements with different levels, the newly inserted node is constrained at the middle of the common edge until the lower-level element is also subdivided.

5. Material Modeling

Yield functions are mathematical descriptions of the states of stresses that will cause yielding. A general form of yield functions is $\varphi (\mathsf{\sigma })=\varphi ({\sigma }_{ij})=A{\overline{\sigma }}^m$ with $\overline{\sigma }$ being the equivalent stress. The plastic strain increment can be obtained by the associated flow rule:

$$\mathrm{d}{\mathsf{\epsilon }}^{\mathrm{p}}=\lambda \frac{\partial \varphi }{\partial \mathsf{\sigma }}.$$

(34)

According to the work-equivalent principle ${\mathsf{\sigma }}^{\mathrm{T}}\mathrm{d}{\mathsf{\epsilon }}^{\mathrm{p}}=\overline{\sigma }\mathrm{d}{\overline{\epsilon }}^{\mathrm{p}}$, in which $\mathrm{d}{\overline{\epsilon }}^{\mathrm{p}}$ is the increment of equivalent plastic strain, and considering that $\overline{\sigma }$ is a first order homogenous function, i.e., $\overline{\sigma }={\mathsf{\sigma }}^{\mathrm{T}}\frac{\partial \overline{\sigma }}{\partial \mathsf{\sigma }}$, we obtain:

$$\mathrm{d}{\mathsf{\epsilon }}^{\mathrm{p}}=\frac{\partial \overline{\sigma }}{\partial \mathsf{\sigma }}\mathrm{d}{\overline{\epsilon }}^{\mathrm{p}}$$

(35)

The combination of Equations (34) and (35) produces:

$$\mathrm{d}{\overline{\epsilon }}^{\mathrm{p}}=\lambda \frac{\mathrm{d}\varphi }{\mathrm{d} \overline{\mathsf{\sigma }}}.$$

(36)

Furtherly, the substitution of Equation (36) into the consistency condition [30] as follows:

$${\left(\frac{\partial \varphi }{\partial \mathsf{\sigma }}\right)}^{\mathrm{T}}\mathrm{d}\mathsf{\sigma }=\frac{\mathrm{d}\varphi }{\mathrm{d}{\overline{\epsilon }}^{\mathrm{p}}}\mathrm{d}{\overline{\epsilon }}^{\mathrm{p}}$$

(37)

leads to:

$${\left(\frac{\partial \varphi }{\partial \mathsf{\sigma }}\right)}^{\mathrm{T}}\mathrm{d}\mathsf{\sigma }=\lambda {\left(\frac{\mathrm{d}\varphi }{\mathrm{d} \overline{\mathsf{\sigma }}}\right)}^2\frac{\mathrm{d} \overline{\mathsf{\sigma }}}{\mathrm{d}{\overline{\epsilon }}^{\mathrm{p}}}=\lambda {\left(Am{\overline{\mathsf{\sigma }}}^{m-1}\right)}^2{E}^{\mathrm{p}}$$

(38)

where plastic modulus $E^{\mathrm{p}}$ relates to Young’s modulus E and tangent modulus $E^{\mathrm{t}}=\frac{\mathrm{d}\overline{\sigma }}{\mathrm{d}\overline{\epsilon }}$ in the form $\frac{1}{E^{\mathrm{t}}}=\frac{1}{E}+\frac{1}{E^{\mathrm{p}}}$.

Based on the classical decomposition of strain into elastic and plastic components, the stress increment is achieved by:

$$\mathrm{d}\mathsf{\sigma }=D^{\mathrm{e}}\mathrm{d}{\mathsf{\epsilon }}^{\mathrm{e}}=D^{\mathrm{e}}(\mathrm{d}\mathsf{\epsilon }-\mathrm{d}{\mathsf{\epsilon }}^{\mathrm{p}})$$

(39)

Substituting Equation (34) into Equation (39) gives:

$$\mathrm{d}\mathsf{\sigma }=D^{\mathrm{e}}(\mathrm{d}\mathsf{\epsilon }-\lambda \frac{\partial \varphi }{\partial \mathsf{\sigma }})$$

(40)

By consolidating Equations (38) and (40), the plastic multiplier $\lambda$ can be obtained as:

$$\lambda =\frac{{\left(\frac{\partial \varphi }{\partial \mathsf{\sigma }}\right)}^{\mathrm{T}}{D}^{\mathrm{e}}\mathrm{d}\mathsf{\epsilon }}{{\left(\frac{\partial \varphi }{\partial \mathsf{\sigma }}\right)}^{\mathrm{T}}{D}^{\mathrm{e}}\frac{\partial \varphi }{\partial \mathsf{\sigma }}+{\left(Am{ \overline{\mathsf{\sigma }}}^{m-1}\right)}^2{E}^{\mathrm{p}}}$$

(41)

Therefore, Equation (40) can be rewritten as $\mathrm{d}\mathsf{\sigma }=D^{\mathrm{ep}}\mathrm{d}\mathsf{\epsilon }$ with the elastoplastic tangent stiffness matrix taking the form:

$$D^{\mathrm{ep}}=D^{\mathrm{e}}-\frac{D^{\mathrm{e}}\frac{\partial \varphi }{\partial \mathsf{\sigma }}{\left(\frac{\partial \varphi }{\partial \mathsf{\sigma }}\right)}^{\mathrm{T}}{D}^{\mathrm{e}}}{{\left(\frac{\partial \varphi }{\partial \mathsf{\sigma }}\right)}^{\mathrm{T}}{D}^{\mathrm{e}}\frac{\partial \varphi }{\partial \mathsf{\sigma }}+{\left(Am{ \overline{\mathsf{\sigma }}}^{m-1}\right)}^2{E}^{\mathrm{p}}}$$

(42)

Since the strain increment in a time step is small enough, $D^{\mathrm{ep}}$ is considered to be a constant matrix in a time step. As shown in Equation (42), the derivative of the yield function is necessary for the calculation of $D^{\mathrm{ep}}$. To describe the anisotropic behavior of metal sheets, many yield functions have been proposed. In the present study, three typical yield functions are programmed to work together with the proposed solid shell to analyze sheet metal forming processes.

5.1. Hill48 Yield Function

Being a well-accepted theory, the yield function by Hill [23] has been a popular choice, particularly for steels. The Hill48 yield function is defined as:

$$\varphi (\mathsf{\sigma })=\overline{F}{\left({\sigma }_{yy}-{\sigma }_{zz}\right)}^2+\overline{G}{\left({\sigma }_{zz}-{\sigma }_{xx}\right)}^2+\overline{H}{\left({\sigma }_{xx}-{\sigma }_{yy}\right)}^2+2\overline{L}{\sigma }_{yz}^2+2\overline{M}{\sigma }_{zx}^2+2\overline{N}{\sigma }_{xy}^2={\overline{\sigma }}^2$$

(43)

The parameters $\overline{F}$, $\overline{G}$, $\overline{H}$, $\overline{L}$, $\overline{M}$ and $\overline{N}$ can be obtained via the Lankford coefficients $r_0$, $r_{45}$ and $r_{90}$ according to the following form:

$$\overline{F}=\frac{r_0}{\left(1+r_0\right)r_{90}} \overline{G}=\frac{1}{1+r_0} \overline{H}=\frac{r_0}{1+r_0} \overline{L}=\overline{M}=\overline{N}=\frac{\left(1+2r_{45}\right)\left(r_0+r_{90}\right)}{2\left(1+r_0\right)r_{90}}$$

(44)

The Hill48 yield function is easy to program since we can take the partial derivatives of Equation (43) in a straightforward way.

5.2. Yld91 Yield Function

The Yld91 yield function suggested by Barlat et al. [24] has also been widely used in sheet metal forming simulations, especially when aluminum sheets are employed. In Yld91, the orthotropic anisotropy is expressed via a matrix involving six independent anisotropy coefficients so that a new stress tensor $\tilde{s}=\overline{C}s=\overline{C}\overline{T}\mathsf{\sigma }=\overline{L}\mathsf{\sigma }$ in the matrix form:

$$\tilde{s}=\left[\begin{array}{c}{\tilde{s}}_{xx}\\ {\tilde{s}}_{yy}\\ {\tilde{s}}_{zz}\\ {\tilde{s}}_{xy}\\ {\tilde{s}}_{yz}\\ {\tilde{s}}_{zx}\end{array}\right]=\frac{1}{3}\left[\begin{array}{cccccc}{C}_2+C_3& -C_3& -C_2& 0& 0& 0\\ -C_3& C_1+C_3& -C_1& 0& 0& 0\\ -C_2& -C_1& C_1+C_2& 0& 0& 0\\ 0& 0& 0& 3C_4& 0& 0\\ 0& 0& 0& 0& 3C_5& 0\\ 0& 0& 0& 0& 0& 3C_6\end{array}\right]\left[\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\sigma }_{zz}\\ {\sigma }_{xy}\\ {\sigma }_{yz}\\ {\sigma }_{zx}\end{array}\right]$$

(45)

is introduced. Due to the transformation from the stress tensor $\mathsf{\sigma }$ to its deviator $s$, the yield function is pressure independent. The principal values of $\tilde{s}$ are the roots of the characteristic equation:

$$P\left({\tilde{S}}_k\right)=-{\tilde{S}}_k^3+3H_1{\tilde{S}}_k^2+3H_2{\tilde{S}}_k+2H_3=0$$

(46)

with the associated 1st, 2nd, and 3rd invariants of $\tilde{s}$ being:

$$\begin{array}{l}{H}_1=\left({\tilde{s}}_{xx}+{\tilde{s}}_{yy}+{\tilde{s}}_{zz}\right)/3\\ H_2=\left({\tilde{s}}_{xy}^2+{\tilde{s}}_{yz}^2+{\tilde{s}}_{zx}^2-{\tilde{s}}_{xx}{\tilde{s}}_{yy}-{\tilde{s}}_{yy}{\tilde{s}}_{zz}-{\tilde{s}}_{zz}{\tilde{s}}_{xx}\right)/3\\ H_3=\left(2{\tilde{s}}_{yz}{\tilde{s}}_{zx}{\tilde{s}}_{xy}+{\tilde{s}}_{xx}{\tilde{s}}_{yy}{\tilde{s}}_{zz}-{\tilde{s}}_{xx}{\tilde{s}}_{yz}^2-{\tilde{s}}_{yy}{\tilde{s}}_{zx}^2-{\tilde{s}}_{zz}{\tilde{s}}_{xy}^2\right)/2\end{array}$$

(47)

The three principal values ${\tilde{S}}_1$, ${\tilde{S}}_2$, and ${\tilde{S}}_3$ are used to express the yield function:

$$\varphi \left(s\right)={\left|{\tilde{S}}_1-{\tilde{S}}_2\right|}^a+{\left|{\tilde{S}}_2-{\tilde{S}}_3\right|}^a+{\left|{\tilde{S}}_3-{\tilde{S}}_1\right|}^a=2{\overline{\sigma }}^a$$

(48)

where $a$ is a constant coefficient being 6 and 8 for body-centered cubic (BCC) and face-centered cubic (FCC) materials, respectively.

By taking the partial derivatives of Equations (45)–(48), the derivative of the yield function, that is:

$$\frac{\partial \varphi }{\partial {\sigma }_{ij}}=\frac{\partial \varphi }{\partial {\tilde{S}}_p}\frac{\partial {\tilde{S}}_p}{\partial H_q}\frac{\partial H_q}{\partial {\tilde{s}}_{rs}}\frac{\partial {\tilde{s}}_{rs}}{\partial {\sigma }_{ij}}$$

(49)

can be worked out and used for the calculation of the tangent stiffness matrix.

5.3. Yld2004-18pYield Function

To characterize the anisotropic behavior of metal sheets more precisely, Barlat et al. [25] developed another yield function denoted as Yld2004-18p:

$$\varphi \left(s\right)={\displaystyle \sum _{i,j}^{1,3}{\left|{\tilde{S}}_i^{\prime }-{\tilde{S}}_j^{″}\right|}^a}=4{\overline{\sigma }}^a$$

(50)

in which two linear transformations are involved, i.e., ${\tilde{s}}^{\prime }={\overline{C}}^{\prime }s={\overline{C}}^{\prime }\overline{T}\mathsf{\sigma }$ and ${\tilde{s}}^{″}={\overline{C}}^{″}s={\overline{C}}^{″}\overline{T}\mathsf{\sigma }$, the linear transformations on the stress deviator being:

$${\overline{C}}^{\prime }=\left[\begin{array}{cccccc}0& -c_{12}^{\prime }& -c_{13}^{\prime }& 0& 0& 0\\ -c_{21}^{\prime }& 0& -c_{23}^{\prime }& 0& 0& 0\\ -c_{31}^{\prime }& -c_{32}^{\prime }& 0& 0& 0& 0\\ 0& 0& 0& c_{44}^{\prime }& 0& 0\\ 0& 0& 0& 0& c_{55}^{\prime }& 0\\ 0& 0& 0& 0& 0& c_{66}^{\prime }\end{array}\right], {\overline{C}}^{″}=\left[\begin{array}{cccccc}0& -c_{12}^{″}& -c_{13}^{″}& 0& 0& 0\\ -c_{21}^{″}& 0& -c_{23}^{″}& 0& 0& 0\\ -c_{31}^{″}& -c_{32}^{″}& 0& 0& 0& 0\\ 0& 0& 0& c_{44}^{″}& 0& 0\\ 0& 0& 0& 0& c_{55}^{″}& 0\\ 0& 0& 0& 0& 0& c_{66}^{″}\end{array}\right]$$

(51)

As compared with Yld91, more anisotropic coefficients are introduced in this yield function. And obviously, Yld2004-18p reduced to Yld91 when the two linear transformations are the same. A little different from that in Yld91, the derivative in Yld2004-18p is written as follows:

$$\frac{\partial \varphi }{\partial {\sigma }_{ij}}=\frac{\partial \varphi }{\partial {\tilde{S}}_p^{\prime }}\frac{\partial {\tilde{S}}_p^{\prime }}{\partial H_q^{\prime }}\frac{\partial H_q^{\prime }}{\partial {\tilde{s}}_{rs}^{\prime }}\frac{\partial {\tilde{s}}_{rs}^{\prime }}{\partial {\sigma }_{ij}}+\frac{\partial \varphi }{\partial {\tilde{S}}_p^{″}}\frac{\partial {\tilde{S}}_p^{″}}{\partial H_q^{″}}\frac{\partial H_q^{″}}{\partial {\tilde{s}}_{rs}^{″}}\frac{\partial {\tilde{s}}_{rs}^{″}}{\partial {\sigma }_{ij}}$$

(52)

6. Numerical Examples

6.1. Deep Drawing of a Square Cup

The deep drawing of a square cup, a benchmark test of the Numisheet 1993 conference [31], is considered in this example. The geometry of the forming tools is sketched in Figure 6, in which a square blank of 150 × 150 × 0.78 mm is clamped by the blank holder and the die [32]. According to the data provided by the organizing committee, the friction coefficient between the blank and the tools is 0.144. Under a constant blank holder force of 16.6 kN, the blank is drawn into the die until the punch reaches its final stroke of 40 mm. A mild steel material is investigated, with Young’s modulus $E=$ 206,000 MPa, Poisson ratio $v=$ 0.3, mean yield stress ${\sigma }_s=$ 173.1 MPa, Lankford coefficients $r_0=$ 1.79, $r_{45}=$ 1.51, $r_{90}=$ 2.27, and mean hardening curve $\overline{\sigma }=565.32{\left(0.007117+{\overline{\epsilon }}^p\right)}^{0.2589}$ MPa.

The proposed solid-shell element with five integration points along the thickness direction was adopted to analyze the drawing process. To validate the performance of the solid-shell element in adaptive mesh subdivision, two simulations were conducted for comparison. In the first simulation, the blank was initially discretized by 35 × 35 solid-shell elements with the adaptive mesh subdivision utilized during the forming process. The highest subdivision level was set as 1. And in the second simulation, the blank was discretized by 70 × 70 elements, with the element number kept constant during the simulation. Aiming at capturing the anisotropic behavior of the metal sheet, the Hill48 yield function was incorporated in the two simulations.

The time costs of these two simulations were 15 min and 10 min, respectively. It is indicated that the proposed adaptive mesh subdivision contributes to higher efficiency. The simulation results, for the first simulation, when the punch reaches its full stroke are shown in Figure 7, in which the element number of the square cup is 3397, much less than the one in the second simulation.

The forming results, including the stress and strain distributions at the final punch stroke, are given in Figure 8. We can find that the material experiences the highest stress in the areas contacting with the punch shoulders. This is because the material in these areas undergoes double-curved bending. Meanwhile, higher blank holder force makes it more difficult for the material in these areas to flow into the die. On the contrary, the material contacting the punch bottom experiences the minimum deformation, and the equivalent strain in this area is approximately zero.

In Figure 8, the draw-in length $D_x$ in the rolling direction, $D_y$ in the transverse direction, and $D_d$ in the diagonal direction are defined. To examine the accuracy of the simulations, the draw-in length in the two simulations were measured and listed in

Table 1. Draw-in results for comparison (units in mm).

	${\mathit{D}}_{\mathit{x}}$	${\mathit{D}}_{\mathit{y}}$	${\mathit{D}}_{\mathit{d}}$
Min. experiment	26.75	26.75	14.60
Max. experiment	29.60	29.58	16.31
35 × 35 mesh with subdivision	27.93	27.73	15.20
70 × 70 mesh	29.22	29.21	16.02

, and compared with experimental results [12]. We can find that even though there is a little difference between the results provided by the two simulations, all the simulation results lie between the minimum and maximum experimental results. This proves that the solid-shell element is suitable for sheet metal forming applications, and the computational stability has not been affected by the adaptive mesh subdivision.

6.2. Earing Prediction for a Cylindrical Cup

To examine the applicability of the constitutive models and evaluate the performance of the solid-shell element in anisotropic problems, a cylindrical cup drawing test was carried out with an AA2090-T3 sheet. The schematic view is shown in Figure 9 [34]. According to the experimental conditions, a constant blank holder force of 22.2 kN and a friction coefficient of 0.1 were employed [34].

The geometric parameters of the blank are: initial thickness $t_0$ = 1.6 mm and diameter $d$ = 158.76 mm. The circular blank was discretized by the proposed solid-shell element with five integration points across the thickness. A quarter of the finite element model is shown in Figure 10, in which the x-axis coincides with the rolling direction. The six-node prismatic elements in the center of the blank were considered to be special hexahedral elements involving coincident nodes. Simulations were conducted with three different yield functions, and the anisotropic coefficients are given in

Table 2. Yield function coefficients of AA2090-T3 [33]. Adapted from Ref. [33] with permission from Elsevier (2021).

Hill48 r-Values
$r_0$		$r_{45}$		$r_{90}$
0.2115		1.5769		0.6923
Yld91 Coefficients
$C_1$		$C_2$		$C_3$		$C_4$	$C_5$	$C_6$	$a$
1.0674		0.8559		1.1296		1.2970	1.0000	1.0000	8
Yld2004-18p Coefficients
$c_{12}^{\prime }$	$c_{13}^{\prime }$	$c_{21}^{\prime }$	$c_{23}^{\prime }$	$c_{31}^{\prime }$	$c_{32}^{\prime }$	$c_{44}^{\prime }$	$c_{55}^{\prime }$	$c_{66}^{\prime }$	$a$
−0.0698	0.9364	0.0791	1.0030	0.5247	1.3631	0.9543	1.0237	1.0690	8
$c_{12}^{″}$	$c_{13}^{″}$	$c_{21}^{″}$	$c_{23}^{″}$	$c_{31}^{″}$	$c_{32}^{″}$	$c_{44}^{″}$	$c_{55}^{″}$	$c_{66}^{″}$
0.9811	0.4767	0.5753	0.8668	1.1450	−0.0792	1.4046	1.0516	1.1471

. The stress–strain curve of the material is $\overline{\sigma }=646{\left(0.025+\overline{\epsilon }\right)}^{0.227}$ MPa.

In Figure 11, the cup height is plotted versus the angle from the rolling direction. It can be observed that the material orthotropy is not completely satisfied in the experimental results, which may be caused by the inexact alignment between the center of the blank and the center of the tools during the drawing process [33]. By comparing the experimental data with the simulation results based on Hill48, Yld91, and Yld2004-18p, we can conclude that the proposed solid-shell element behaves well in all the simulations. A closer look shows that Hill48 exaggerates the earing phenomenon. The result calculated by Yld91 is in conformity with the experimental one, except that the small ears at 0° and 180° are not exactly predicted. From the viewpoint of earing prediction, the best agreement with the experimental result is obtained by Yld2004-18p.

6.3. Drawing of an Automobile Component

The drawing process of a complex automobile component is shown in Figure 12 and was analyzed to examine the accuracy and stability of the proposed element. The initial size of the blank is 270 × 550 × 1.5 mm. The finite element model of the drawing simulation is displayed in Figure 13, in which the blank is discretized by 22,246 solid-shell elements with five integration points in the thickness direction. At the beginning of the drawing process, the blank is located on the blank holder, and the die moves in the -z direction at a constant speed of 5 m/s. When the blank holder moves along with the die, the blank holder force increases linearly with the displacement. The initial blank holder force is 558.4 kN, and the final blank holder force at the full stroke is 764.9 kN. According to the many experiments conducted in the automobile enterprise, the lubrication condition between the blank and the tools in this example should yield a friction coefficient of 0.15.

The AA5754 material is used for the drawing process, with the x-axis in Figure 13 being the rolling direction. By uniaxial tensile tests in 0°, 45°, and 90° from the rolling direction, we obtained the following material parameters: Young’s modulus $E=$ 70,000 MPa, Poisson ratio $v=$ 0.3, hardening curve $\overline{\sigma }=482.5{(0.007117+{\overline{\epsilon }}^p)}^{0.324}$ MPa. The yield stresses ${\sigma }_0$, ${\sigma }_{45}$, and ${\sigma }_{90}$ obtained by the uniaxial tensile tests, along with the balanced biaxial yield stress ${\sigma }_b$ measured form a bulge test, were used to fit the coefficients $C_{1~4}$ of Yld91 for the AA5754 blank. The experimental data and the Yld91 coefficients are listed in

Table 3. Anisotropy coefficients of AA5754. Adapted from Ref. [33] with permission from Elsevier (2021).

Experimental Data (MPa)
${\sigma }_0$	${\sigma }_{45}$	${\sigma }_{90}$	${\sigma }_b$
99.0	100.5	102.5	98.9
Yld91 coefficients
$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$C_6$	$a$
0.9669	1.0342	0.9648	0.9797	1.0000	1.0000	8

.

The adaptive mesh subdivision with the highest subdivision level being 2 was adopted. The simulation was conducted on an 8-core PC, and the computation time was 7.2 h. The finite element model of the final part is shown in Figure 14, in which the in-plane size of the smallest element is 1.41 mm, and the element number is 193,804. According to Figure 14, the adaptive mesh subdivision brings about finer mesh for some local areas, which makes the element number as low as possible.

The major strain and the minor strain in the experiment were obtained by the optical strain measurement system ARGUS (GOM). Specifically, a hexagonal dot pattern was marked on the blank using electro-chemical etching before the drawing process. After the drawing process, the dots on the component were recorded by a handheld ARGUS camera (Argus, Michigan, MI, USA) from different viewing angles. Finally, the 2D coordinates of all the dots were mathematically derived and recalculated to 3D coordinates using photogrammetry principles. Thus, the strain values can be determined by evaluating the relative distance between the dots in combination with a local plane strain tensor computation. The measured results are given in Figure 15 and Figure 16 in comparison with the simulation results. It should be noted that the flange areas at the left and right ends are not shown in Figure 15b and Figure 16b, since the strain of these areas cannot be measured precisely by ARGUS. For this reason, the minimum values of the legends in the simulation and the experiment differ obviously. Some elements in Figure 15a and Figure 16a are picked out, with their strain values as compared with the measured ones in Figure 15b and Figure 16b. It can be found that the calculated strain distributions are overall in good conformity with the experimental results. The major strain values at the picked points in Figure 15 are plotted in Figure 17. It can be found that the maximum major strain is about 0.15, which exists at Point 3 and Point 7, indicating that when the die reaches its full stroke, large deformation takes place at these two areas since the material feeding resistance is considerable. Among the 14 points, only the errors at Points 1 and 13 are greater than 30%, and the errors at most of the points are less than 10%.

7. Conclusions

In this study, we applied a recently proposed solid-shell element to explicit analyses of sheet metal forming. In the deep drawing of a square cup, the solid-shell element achieves excellent agreement with the experimental draw-in results. It also proves that the proposed adaptive mesh subdivision can dramatically reduce the computational cost without obviously affecting simulation accuracy. Acting together with different constitutive laws involving Hill48, Yld91, and Yld2004-18p yield functions, a good correlation to the experimental results was obtained in predicting cup height. Among the three yield functions, Yld2004-18p demonstrated its superior performance in predicting the earing contour of the fully drawn cup. To further validate the accuracy in describing strain over the entire surface, the drawing process of an automobile component was computed, and again the accuracy of the proposed element and the robustness of the adaptive mesh subdivision were demonstrated. Altogether, the proposed solid-shell element is well suited for sheet metal forming simulations.